Trees with exactly three main eigenvalues
arXiv:2607.13577
The paper classifies all trees of diameter five that have exactly three main eigenvalues, showing they are either symmetric trees T_r(a) or belong to a parametric family defined by arithmetic divisibility conditions, and also constructs an infinite family with unbounded diameter.
Abstract
An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector. Introduced by CvetkoviÄ in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood. However, the classification of graphs with precisely three main eigenvalues remains a challenging open problem in spectral graph theory. This paper provides a complete classification of all trees of diameter 5 with exactly three main eigenvalues. Using equitable partitions, the spectral condition reduces to the unique solvability of linear systems over the rationals, leading to Diophantine equations involving branch lengths and pendant counts. We prove that every such tree is isomorphic either to a symmetric tree or to a member of a parametric family determined by arithmetic divisibility conditions. We also construct an infinite family of such trees with unbounded diameter.
18 pages